Theorem reximdva

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of Margaris p. 90. (Contributed by NM, 22-May-1999)

	Ref	Expression
Hypothesis	ralimdva.1	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \to \chi \right)$$
Assertion	reximdva	$$⊢\phi \to \left(\exists x\in A\psi \to \exists x\in A\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ralimdva.1	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \to \chi \right)$$
2	1	ex	$$⊢\phi \to \left(x\in A\to \left(\psi \to \chi \right)\right)$$
3	2	reximdvai	$$⊢\phi \to \left(\exists x\in A\psi \to \exists x\in A\chi \right)$$